Validation of mathematical model for CZ process using small-scale laboratory crystal growth furnace

Bergfelds, Kristaps; Sabanskis, Andrejs; Virbulis, J.

doi:10.1088/1757-899x/355/1/012004

Cited by 4 publications

(12 citation statements)

References 3 publications

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“…where λ c is the thermal conductivity and V is the crystal pull rate, which has only a vertical component V z = V. While the melt flow was not simulated, to correctly capture the crystallization interface shapes its influence was approximated by (i) homogeneous but time-dependent melt thermal conductivity λ m (t) and (ii) redistribution of the heat flux densities along the crystallization interface [30]. For the crystal λ c (T) = 98.89 − 9.43 × 10 −2 T + 2.89 × 10 −5 T 2 (in W/m/K) was used; emissivities were 0.64 and 0.30 for the solid and liquid Si, respectively.…”

Section: Heat Transfer and Phase Boundariesmentioning

confidence: 99%

“…While the heat transfer in the crystal and q c is straightforward to calculate, q m can be strongly influenced by the melt flow, which is not solved explicitly. Its influence is included in the model by introducing the heat flux density correction q corr [30], which describes redistribution of heat sources along the interface, calculated based on the experimental interface shapes. The values of q corr (r) are obtained at certain positions in the crystal and are linearly interpolated according to the time-dependent crystal length.…”

Section: Thermal Stressesmentioning

confidence: 99%

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Evaluation of the Performance of Published Point Defect Parameter Sets in Cone and Body Phase of a 300 mm Czochralski Silicon Crystal

Sabanskis

Plāte²,

Sattler³

et al. 2021

Crystals

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Prediction and adjustment of point defect (vacancies and self-interstitials) distribution in silicon crystals is of utmost importance for microelectronic applications. The simulation of growth processes is widely applied for process development and quite a few different sets of point defect parameters have been proposed. In this paper the transient temperature, thermal stress and point defect distributions are simulated for 300 mm Czochralski growth of the whole crystal including cone and cylindrical growth phases. Simulations with 12 different published point defect parameter sets are compared to the experimentally measured interstitial–vacancy boundary. The results are evaluated for standard and adjusted parameter sets and generally the best agreement in the whole crystal is found for models considering the effect of thermal stress on the equilibrium point defect concentration.

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Section: Heat Transfer and Phase Boundariesmentioning

confidence: 99%

Section: Thermal Stressesmentioning

confidence: 99%

Evaluation of the Performance of Published Point Defect Parameter Sets in Cone and Body Phase of a 300 mm Czochralski Silicon Crystal

Sabanskis

Plāte²,

Sattler³

et al. 2021

Crystals

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“…Laminar convection of different origins was studied: thermal gravity, thermocapillary and thermal gravity capillary [10][11][12][13][14][15].…”

Section: Numerical Study Problem Statementmentioning

confidence: 99%

“…if the temperature control system does not produce disturbance). Determination of theoretical stability limits is another problem of physical and numerical simulation [1,[13][14][15][16].…”

Section: Introductionmentioning

confidence: 99%

Hydrodynamics and heat exchange of crystal pulling from melts. Part II: Numerical study of free convection mode

Бердников¹

2021

MoEM

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This work is a brief overview of numerical study results for hydrodynamics and convective heat exchange for the classic Czochralski technique obtained at the Institute of Thermophysics, Siberian Branch of the Russian Academy of Sciences. Flow structure evolution has been compared for melts with Prandtl numbers Pr = 0.05, 16, 50 and 2700. The regularities of local and integral convective heat exchange in the crucible/melt/crystal system for thermal gravity, thermocapillary and heat-induced gravity capillary convection have been studied. The calculations have been carried out using the finite and compact difference methods. This work is a continuation of an article: Berdnikov VS (2019) Hydrodynamics and heat exchange of crystal pulling from melts. Part I: Experimental studies of free convection mode. Modern Electronic Materials 5(3): 91–100. https://doi.org/10.3897/j.moem.5.3.46647

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“…Optimization of single crystal growth process modes requires a combined effort including physical and mathematical simulation of the growth processes as well as analysis of the homogeneity of the as-grown crystals. Numerical simulation of single crystal growth processes for the Czochralski technique is currently the focus of intense research for various growth chamber design options [6][7][8][9][10][11][12], yet there are no works that would combine numerical simulation of the hydrodynamic growth conditions with electrical resistivity simulation and measurement in growing crystals. It has been believed that, once a flat crystallization front is achieved, the main cause of radial inhomogeneities is the crystallographic symmetry of planes in the growing crystal.…”

Section: Introductionmentioning

confidence: 99%

Possible causes of electrical resistivity distribution inhomogeneity in Czochralski grown single crystal silicon

Kobeleva¹,

Anfimov²,

Бердников³

et al. 2019

MoEM

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Electrical resistivity distribution maps have been constructed for single crystal silicon wafers cut out of different parts of Czochralski grown ingots. The general inhomogeneity of the wafers has proven to be relatively high, the resistivity scatter reaching 1–3 %. Two electrical resistivity distribution inhomogeneity types have been revealed: azimuthal and radial. Experiments have been carried out for crystal growth from transparent simulating fluids with hydrodynamic and thermophysical parameters close to those for Czochralski growth of silicon single crystals. We show that a possible cause of azimuthal electrical resistivity distribution inhomogeneity is the swirl-like structure of the melt under the crystallization front (CF), while a possible cause of radial electrical resistivity distribution inhomogeneity is the CF curvature. In a specific range of the Grashof, Marangoni and Reynolds numbers which depend on the ratio of melt height and growing crystal radius, a system of well-developed radially oriented swirls may emerge under the rotating CF. In the absence of such swirls the melt is displaced from under the crystallization front in a homogeneous manner to form thermal and concentration boundary layers which are homogeneous in azimuthal direction but have clear radial inhomogeneity. Once swirls emerge the melt is displaced from the center to the periphery, and simultaneous fluid motion in azimuthal direction occurs. The overall melt motion becomes helical as a result. The number of swirls (two to ten) agrees with the number of azimuthally directed electrical resistivity distribution inhomogeneities observed in the experiments. Comparison of numerical simulation results in a wide range of Prandtl numbers with the experimental data suggests that the phenomena observed in transparent fluids are universal and can be used for theoretical interpretation of imperfections in silicon single crystals.

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Validation of mathematical model for CZ process using small-scale laboratory crystal growth furnace

Cited by 4 publications

References 3 publications

Evaluation of the Performance of Published Point Defect Parameter Sets in Cone and Body Phase of a 300 mm Czochralski Silicon Crystal

Evaluation of the Performance of Published Point Defect Parameter Sets in Cone and Body Phase of a 300 mm Czochralski Silicon Crystal

Hydrodynamics and heat exchange of crystal pulling from melts. Part II: Numerical study of free convection mode

Possible causes of electrical resistivity distribution inhomogeneity in Czochralski grown single crystal silicon

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Validation of mathematical model for CZ process using small-scale laboratory crystal growth furnace

Cited by 4 publications

References 3 publications

Evaluation of the Performance of Published Point Defect Parameter Sets in Cone and Body Phase of a 300 mm Czochralski Silicon Crystal

Evaluation of the Performance of Published Point Defect Parameter Sets in Cone and Body Phase of a 300 mm Czochralski Silicon Crystal

﻿Hydrodynamics and heat exchange of crystal pulling from melts. Part II: Numerical study of free convection mode

Possible causes of electrical resistivity distribution inhomogeneity in Czochralski grown single crystal silicon

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Hydrodynamics and heat exchange of crystal pulling from melts. Part II: Numerical study of free convection mode